In mathematics, the Conway groups Co₁, Co₂, and Co₃ are three sporadic groups discovered by John Horton Conway. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ... See John B. Conway for the functional analyst. ...

All are closely related to the Leech lattice Λ. The largest, Co₁, of order In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ...

4,157,776,806,543,360,000,

is obtained by dividing the automorphism group of Λ by its center, which consists of the scalar matrices ±1. The groups Co₂ (of order 42,305,421,312,000) and Co₃ (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of length 2 and a vector of √6 respectively. As the scalar −1 fixes no non-zero vector, we can regard these two groups as subgroups of Co₁. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if...

Other sporadic groups

The groups Co₂ and Co₃ both contain the McLaughlin group McL (of order 898,128,000) and the Higman-Sims group (of order 44,352,000), which can be described as the pointwise stabilizers of a In mathematics, the Higman-Sims group is a finite sporadic simple group of order 44352000. ...

2-2-√6 triangle

and a

2-√6-√6 triangle,

respectively. Identifying R²⁴ with C¹² and Λ with

Z[e^2πi/3]¹²,

the resulting automorphism group, i.e., the group of Leech lattice automorphisms preserving the complex structure, when divided by the 6-element group of complex scalar matrices, gives the Suzuki group Suz (of order 448,345,497,600). In mathematics, a complex structure on a real vector space V is a real linear transformation J : V → V such that J2 = −idV. Here J2 means J composed with itself and idV is the identity map on V. That is, the effect of applying J twice is the same as...

A similar construction gives the Hall-Janko group J₂ (of order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

References

• Conway, J. H.: A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398-400.
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press, 1985, ISBN 0-19-853199-0